A389127 a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(n+k,n-3*k).

1, 1, 1, 4, 21, 76, 226, 680, 2269, 7915, 27136, 91246, 307374, 1047619, 3595866, 12352159, 42415949, 145850821, 502698331, 1735954780, 6001926436, 20770641161, 71951921108, 249507459224, 866031978142, 3008426000926, 10458378159451, 36382333622116

Offset

0,4

Comments

Binomial transform of A389125.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A389125(k).

a(n) = [x^n] (1 + x + x^3 * (1 + x)^2)^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / (1 + x + x^3 * (1 + x)^2) ). See A389131.

Mathematica

Table[Sum[Binomial[n, k]Binomial[n+k, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\3, binomial(n, k)*binomial(n+k, n-3*k));
(Magma) [&+[Binomial(n, k) * Binomial(n+k, n-3*k): k in [0..Floor(n/3)]]: n in [0..30]]; // Vincenzo Librandi, Sep 26 2025

Crossrefs

Cf. A389125, A389131.

Sequence in context: A034960 A240372 A157493 * A192429 A354172 A280434

Adjacent sequences: A389124 A389125 A389126 * A389128 A389129 A389130

Keyword

nonn

Author

Seiichi Manyama, Sep 24 2025

Status

approved